AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1

Percentage Accurate: 59.8% → 98.6%

Time: 12.2s

Alternatives: 14

Speedup: 2.3×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
}

Python

def code(x, y, z, t, a, b):
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]

Initial Program: 59.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))

C

double code(double x, double y, double z, double t, double a, double b) {
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
}

Python

def code(x, y, z, t, a, b):
	return ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y)

Julia

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b)) / Float64(Float64(x + t) + y))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = ((((x + y) * z) + ((t + y) * a)) - (y * b)) / ((x + t) + y);
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision] / N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 98.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(t + x\right)\\ \left(\frac{z - b}{\frac{t_1}{y}} + \frac{a}{\frac{t_1}{y + t}}\right) + \frac{z}{\frac{t_1}{x}} \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ t x))))
   (+ (+ (/ (- z b) (/ t_1 y)) (/ a (/ t_1 (+ y t)))) (/ z (/ t_1 x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (t + x);
	return (((z - b) / (t_1 / y)) + (a / (t_1 / (y + t)))) + (z / (t_1 / x));
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    t_1 = y + (t + x)
    code = (((z - b) / (t_1 / y)) + (a / (t_1 / (y + t)))) + (z / (t_1 / x))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (t + x);
	return (((z - b) / (t_1 / y)) + (a / (t_1 / (y + t)))) + (z / (t_1 / x));
}

Python

def code(x, y, z, t, a, b):
	t_1 = y + (t + x)
	return (((z - b) / (t_1 / y)) + (a / (t_1 / (y + t)))) + (z / (t_1 / x))

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(t + x))
	return Float64(Float64(Float64(Float64(z - b) / Float64(t_1 / y)) + Float64(a / Float64(t_1 / Float64(y + t)))) + Float64(z / Float64(t_1 / x)))
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	t_1 = y + (t + x);
	tmp = (((z - b) / (t_1 / y)) + (a / (t_1 / (y + t)))) + (z / (t_1 / x));
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(t + x), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(N[(z - b), $MachinePrecision] / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] + N[(a / N[(t$95$1 / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z / N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 58.0%

   \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
2. Step-by-step derivation

3. Simplified 58.3%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, z, \mathsf{fma}\left(y, a - \left(b - z\right), t \cdot a\right)\right)}{x + \left(y + t\right)}} \]

4. Taylor expanded in a around inf 57.9%

   \[\leadsto \color{blue}{\frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \left(\frac{z \cdot x}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right)} \]
5. Step-by-step derivation

   1. +-commutative 57.9%

      \[\leadsto \frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \color{blue}{\left(\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)} + \frac{z \cdot x}{y + \left(t + x\right)}\right)} \]

   2. associate-+r+ 57.9%

      \[\leadsto \color{blue}{\left(\frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right) + \frac{z \cdot x}{y + \left(t + x\right)}} \]

   3. associate-/l* 67.8%

      \[\leadsto \left(\color{blue}{\frac{z - b}{\frac{y + \left(t + x\right)}{y}}} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right) + \frac{z \cdot x}{y + \left(t + x\right)} \]

   4. associate-/l* 82.9%

      \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \color{blue}{\frac{a}{\frac{y + \left(t + x\right)}{y + t}}}\right) + \frac{z \cdot x}{y + \left(t + x\right)} \]

   5. associate-/l* 98.8%

      \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{a}{\frac{y + \left(t + x\right)}{y + t}}\right) + \color{blue}{\frac{z}{\frac{y + \left(t + x\right)}{x}}} \]

6. Simplified 98.8%

   \[\leadsto \color{blue}{\left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{a}{\frac{y + \left(t + x\right)}{y + t}}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}}} \]

7. Final simplification 98.8%

   \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{a}{\frac{y + \left(t + x\right)}{y + t}}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}} \]

Alternative 2: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(y + x\right)\\ t_2 := y + \left(t + x\right)\\ t_3 := \frac{z - b}{\frac{t_2}{y}}\\ t_4 := a \cdot \left(y + t\right)\\ t_5 := \frac{\left(t_4 + t_1\right) - b \cdot y}{t_2}\\ t_6 := \frac{z}{\frac{t_2}{x}}\\ \mathbf{if}\;t_5 \leq -\infty:\\ \;\;\;\;t_6 + \left(t_3 + \frac{y}{\frac{y + x}{a}}\right)\\ \mathbf{elif}\;t_5 \leq 10^{+249}:\\ \;\;\;\;\frac{t_1}{t_2} + \frac{t_4 - b \cdot y}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_6 + \left(t_3 + a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (* z (+ y x)))
        (t_2 (+ y (+ t x)))
        (t_3 (/ (- z b) (/ t_2 y)))
        (t_4 (* a (+ y t)))
        (t_5 (/ (- (+ t_4 t_1) (* b y)) t_2))
        (t_6 (/ z (/ t_2 x))))
   (if (<= t_5 (- INFINITY))
     (+ t_6 (+ t_3 (/ y (/ (+ y x) a))))
     (if (<= t_5 1e+249)
       (+ (/ t_1 t_2) (/ (- t_4 (* b y)) t_2))
       (+ t_6 (+ t_3 a))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (y + x);
	double t_2 = y + (t + x);
	double t_3 = (z - b) / (t_2 / y);
	double t_4 = a * (y + t);
	double t_5 = ((t_4 + t_1) - (b * y)) / t_2;
	double t_6 = z / (t_2 / x);
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = t_6 + (t_3 + (y / ((y + x) / a)));
	} else if (t_5 <= 1e+249) {
		tmp = (t_1 / t_2) + ((t_4 - (b * y)) / t_2);
	} else {
		tmp = t_6 + (t_3 + a);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z * (y + x);
	double t_2 = y + (t + x);
	double t_3 = (z - b) / (t_2 / y);
	double t_4 = a * (y + t);
	double t_5 = ((t_4 + t_1) - (b * y)) / t_2;
	double t_6 = z / (t_2 / x);
	double tmp;
	if (t_5 <= -Double.POSITIVE_INFINITY) {
		tmp = t_6 + (t_3 + (y / ((y + x) / a)));
	} else if (t_5 <= 1e+249) {
		tmp = (t_1 / t_2) + ((t_4 - (b * y)) / t_2);
	} else {
		tmp = t_6 + (t_3 + a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = z * (y + x)
	t_2 = y + (t + x)
	t_3 = (z - b) / (t_2 / y)
	t_4 = a * (y + t)
	t_5 = ((t_4 + t_1) - (b * y)) / t_2
	t_6 = z / (t_2 / x)
	tmp = 0
	if t_5 <= -math.inf:
		tmp = t_6 + (t_3 + (y / ((y + x) / a)))
	elif t_5 <= 1e+249:
		tmp = (t_1 / t_2) + ((t_4 - (b * y)) / t_2)
	else:
		tmp = t_6 + (t_3 + a)
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z * Float64(y + x))
	t_2 = Float64(y + Float64(t + x))
	t_3 = Float64(Float64(z - b) / Float64(t_2 / y))
	t_4 = Float64(a * Float64(y + t))
	t_5 = Float64(Float64(Float64(t_4 + t_1) - Float64(b * y)) / t_2)
	t_6 = Float64(z / Float64(t_2 / x))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(t_6 + Float64(t_3 + Float64(y / Float64(Float64(y + x) / a))));
	elseif (t_5 <= 1e+249)
		tmp = Float64(Float64(t_1 / t_2) + Float64(Float64(t_4 - Float64(b * y)) / t_2));
	else
		tmp = Float64(t_6 + Float64(t_3 + a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = z * (y + x);
	t_2 = y + (t + x);
	t_3 = (z - b) / (t_2 / y);
	t_4 = a * (y + t);
	t_5 = ((t_4 + t_1) - (b * y)) / t_2;
	t_6 = z / (t_2 / x);
	tmp = 0.0;
	if (t_5 <= -Inf)
		tmp = t_6 + (t_3 + (y / ((y + x) / a)));
	elseif (t_5 <= 1e+249)
		tmp = (t_1 / t_2) + ((t_4 - (b * y)) / t_2);
	else
		tmp = t_6 + (t_3 + a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y + N[(t + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(z - b), $MachinePrecision] / N[(t$95$2 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(t$95$4 + t$95$1), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$6 = N[(z / N[(t$95$2 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(t$95$6 + N[(t$95$3 + N[(y / N[(N[(y + x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 1e+249], N[(N[(t$95$1 / t$95$2), $MachinePrecision] + N[(N[(t$95$4 - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(t$95$6 + N[(t$95$3 + a), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -inf.0

   1. Initial program 5.9%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Step-by-step derivation

   3. Simplified 5.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, z, \mathsf{fma}\left(y, a - \left(b - z\right), t \cdot a\right)\right)}{x + \left(y + t\right)}} \]

   4. Taylor expanded in a around inf 5.3%

      \[\leadsto \color{blue}{\frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \left(\frac{z \cdot x}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 5.3%

         \[\leadsto \frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \color{blue}{\left(\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)} + \frac{z \cdot x}{y + \left(t + x\right)}\right)} \]

      2. associate-+r+ 5.3%

         \[\leadsto \color{blue}{\left(\frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right) + \frac{z \cdot x}{y + \left(t + x\right)}} \]

      3. associate-/l* 34.5%

         \[\leadsto \left(\color{blue}{\frac{z - b}{\frac{y + \left(t + x\right)}{y}}} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right) + \frac{z \cdot x}{y + \left(t + x\right)} \]

      4. associate-/l* 65.3%

         \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \color{blue}{\frac{a}{\frac{y + \left(t + x\right)}{y + t}}}\right) + \frac{z \cdot x}{y + \left(t + x\right)} \]

      5. associate-/l* 100.0%

         \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{a}{\frac{y + \left(t + x\right)}{y + t}}\right) + \color{blue}{\frac{z}{\frac{y + \left(t + x\right)}{x}}} \]

   6. Simplified 100.0%

      \[\leadsto \color{blue}{\left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{a}{\frac{y + \left(t + x\right)}{y + t}}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}}} \]

   7. Taylor expanded in t around 0 67.2%

      \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \color{blue}{\frac{a \cdot y}{y + x}}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}} \]
   8. Step-by-step derivation

      1. *-commutative 67.2%

         \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{\color{blue}{y \cdot a}}{y + x}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}} \]

      2. associate-/l* 92.3%

         \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \color{blue}{\frac{y}{\frac{y + x}{a}}}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}} \]

   9. Simplified 92.3%

      \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \color{blue}{\frac{y}{\frac{y + x}{a}}}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}} \]

   if -inf.0 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999992e248

   1. Initial program 99.1%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in z around inf 99.1%

      \[\leadsto \color{blue}{\left(\frac{\left(y + x\right) \cdot z}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right) - \frac{y \cdot b}{y + \left(t + x\right)}} \]
   3. Step-by-step derivation

      1. associate--l+ 99.1%

         \[\leadsto \color{blue}{\frac{\left(y + x\right) \cdot z}{y + \left(t + x\right)} + \left(\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)} - \frac{y \cdot b}{y + \left(t + x\right)}\right)} \]

      2. *-commutative 99.1%

         \[\leadsto \frac{\color{blue}{z \cdot \left(y + x\right)}}{y + \left(t + x\right)} + \left(\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)} - \frac{y \cdot b}{y + \left(t + x\right)}\right) \]

      3. div-sub 99.1%

         \[\leadsto \frac{z \cdot \left(y + x\right)}{y + \left(t + x\right)} + \color{blue}{\frac{a \cdot \left(y + t\right) - y \cdot b}{y + \left(t + x\right)}} \]

   4. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{z \cdot \left(y + x\right)}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right) - y \cdot b}{y + \left(t + x\right)}} \]

   if 9.9999999999999992e248 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 9.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Step-by-step derivation

   3. Simplified 10.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, z, \mathsf{fma}\left(y, a - \left(b - z\right), t \cdot a\right)\right)}{x + \left(y + t\right)}} \]

   4. Taylor expanded in a around inf 9.1%

      \[\leadsto \color{blue}{\frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \left(\frac{z \cdot x}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 9.1%

         \[\leadsto \frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \color{blue}{\left(\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)} + \frac{z \cdot x}{y + \left(t + x\right)}\right)} \]

      2. associate-+r+ 9.1%

         \[\leadsto \color{blue}{\left(\frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right) + \frac{z \cdot x}{y + \left(t + x\right)}} \]

      3. associate-/l* 29.7%

         \[\leadsto \left(\color{blue}{\frac{z - b}{\frac{y + \left(t + x\right)}{y}}} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right) + \frac{z \cdot x}{y + \left(t + x\right)} \]

      4. associate-/l* 65.7%

         \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \color{blue}{\frac{a}{\frac{y + \left(t + x\right)}{y + t}}}\right) + \frac{z \cdot x}{y + \left(t + x\right)} \]

      5. associate-/l* 99.9%

         \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{a}{\frac{y + \left(t + x\right)}{y + t}}\right) + \color{blue}{\frac{z}{\frac{y + \left(t + x\right)}{x}}} \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{a}{\frac{y + \left(t + x\right)}{y + t}}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}}} \]

   7. Taylor expanded in y around inf 86.9%

      \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \color{blue}{a}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(y + x\right)\right) - b \cdot y}{y + \left(t + x\right)} \leq -\infty:\\ \;\;\;\;\frac{z}{\frac{y + \left(t + x\right)}{x}} + \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{y}{\frac{y + x}{a}}\right)\\ \mathbf{elif}\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(y + x\right)\right) - b \cdot y}{y + \left(t + x\right)} \leq 10^{+249}:\\ \;\;\;\;\frac{z \cdot \left(y + x\right)}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right) - b \cdot y}{y + \left(t + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{y + \left(t + x\right)}{x}} + \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + a\right)\\ \end{array} \]

Alternative 3: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(t + x\right)\\ t_2 := \frac{\left(a \cdot \left(y + t\right) + z \cdot \left(y + x\right)\right) - b \cdot y}{t_1}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+124} \lor \neg \left(t_2 \leq 10^{+249}\right):\\ \;\;\;\;\frac{z}{\frac{t_1}{x}} + \left(\frac{z - b}{\frac{t_1}{y}} + a\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ t x)))
        (t_2 (/ (- (+ (* a (+ y t)) (* z (+ y x))) (* b y)) t_1)))
   (if (or (<= t_2 -5e+124) (not (<= t_2 1e+249)))
     (+ (/ z (/ t_1 x)) (+ (/ (- z b) (/ t_1 y)) a))
     t_2)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (t + x);
	double t_2 = (((a * (y + t)) + (z * (y + x))) - (b * y)) / t_1;
	double tmp;
	if ((t_2 <= -5e+124) || !(t_2 <= 1e+249)) {
		tmp = (z / (t_1 / x)) + (((z - b) / (t_1 / y)) + a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y + (t + x)
    t_2 = (((a * (y + t)) + (z * (y + x))) - (b * y)) / t_1
    if ((t_2 <= (-5d+124)) .or. (.not. (t_2 <= 1d+249))) then
        tmp = (z / (t_1 / x)) + (((z - b) / (t_1 / y)) + a)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (t + x);
	double t_2 = (((a * (y + t)) + (z * (y + x))) - (b * y)) / t_1;
	double tmp;
	if ((t_2 <= -5e+124) || !(t_2 <= 1e+249)) {
		tmp = (z / (t_1 / x)) + (((z - b) / (t_1 / y)) + a);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y + (t + x)
	t_2 = (((a * (y + t)) + (z * (y + x))) - (b * y)) / t_1
	tmp = 0
	if (t_2 <= -5e+124) or not (t_2 <= 1e+249):
		tmp = (z / (t_1 / x)) + (((z - b) / (t_1 / y)) + a)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(t + x))
	t_2 = Float64(Float64(Float64(Float64(a * Float64(y + t)) + Float64(z * Float64(y + x))) - Float64(b * y)) / t_1)
	tmp = 0.0
	if ((t_2 <= -5e+124) || !(t_2 <= 1e+249))
		tmp = Float64(Float64(z / Float64(t_1 / x)) + Float64(Float64(Float64(z - b) / Float64(t_1 / y)) + a));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y + (t + x);
	t_2 = (((a * (y + t)) + (z * (y + x))) - (b * y)) / t_1;
	tmp = 0.0;
	if ((t_2 <= -5e+124) || ~((t_2 <= 1e+249)))
		tmp = (z / (t_1 / x)) + (((z - b) / (t_1 / y)) + a);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(t + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -5e+124], N[Not[LessEqual[t$95$2, 1e+249]], $MachinePrecision]], N[(N[(z / N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z - b), $MachinePrecision] / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], t$95$2]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -4.9999999999999996e124 or 9.9999999999999992e248 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 24.6%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Step-by-step derivation

   3. Simplified 25.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, z, \mathsf{fma}\left(y, a - \left(b - z\right), t \cdot a\right)\right)}{x + \left(y + t\right)}} \]

   4. Taylor expanded in a around inf 24.5%

      \[\leadsto \color{blue}{\frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \left(\frac{z \cdot x}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 24.5%

         \[\leadsto \frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \color{blue}{\left(\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)} + \frac{z \cdot x}{y + \left(t + x\right)}\right)} \]

      2. associate-+r+ 24.5%

         \[\leadsto \color{blue}{\left(\frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right) + \frac{z \cdot x}{y + \left(t + x\right)}} \]

      3. associate-/l* 44.3%

         \[\leadsto \left(\color{blue}{\frac{z - b}{\frac{y + \left(t + x\right)}{y}}} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right) + \frac{z \cdot x}{y + \left(t + x\right)} \]

      4. associate-/l* 71.8%

         \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \color{blue}{\frac{a}{\frac{y + \left(t + x\right)}{y + t}}}\right) + \frac{z \cdot x}{y + \left(t + x\right)} \]

      5. associate-/l* 99.9%

         \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{a}{\frac{y + \left(t + x\right)}{y + t}}\right) + \color{blue}{\frac{z}{\frac{y + \left(t + x\right)}{x}}} \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{a}{\frac{y + \left(t + x\right)}{y + t}}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}}} \]

   7. Taylor expanded in y around inf 89.5%

      \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \color{blue}{a}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}} \]

   if -4.9999999999999996e124 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999992e248

   1. Initial program 98.9%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(y + x\right)\right) - b \cdot y}{y + \left(t + x\right)} \leq -5 \cdot 10^{+124} \lor \neg \left(\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(y + x\right)\right) - b \cdot y}{y + \left(t + x\right)} \leq 10^{+249}\right):\\ \;\;\;\;\frac{z}{\frac{y + \left(t + x\right)}{x}} + \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(y + x\right)\right) - b \cdot y}{y + \left(t + x\right)}\\ \end{array} \]

Alternative 4: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + \left(t + x\right)\\ t_2 := a \cdot \left(y + t\right)\\ t_3 := \frac{\left(t_2 + z \cdot \left(y + x\right)\right) - b \cdot y}{t_1}\\ \mathbf{if}\;t_3 \leq -5 \cdot 10^{+124} \lor \neg \left(t_3 \leq 10^{+249}\right):\\ \;\;\;\;\frac{z}{\frac{t_1}{x}} + \left(\frac{z - b}{\frac{t_1}{y}} + a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t_2 + \left(z \cdot y + z \cdot x\right)\right) - b \cdot y}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (+ t x)))
        (t_2 (* a (+ y t)))
        (t_3 (/ (- (+ t_2 (* z (+ y x))) (* b y)) t_1)))
   (if (or (<= t_3 -5e+124) (not (<= t_3 1e+249)))
     (+ (/ z (/ t_1 x)) (+ (/ (- z b) (/ t_1 y)) a))
     (/ (- (+ t_2 (+ (* z y) (* z x))) (* b y)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (t + x);
	double t_2 = a * (y + t);
	double t_3 = ((t_2 + (z * (y + x))) - (b * y)) / t_1;
	double tmp;
	if ((t_3 <= -5e+124) || !(t_3 <= 1e+249)) {
		tmp = (z / (t_1 / x)) + (((z - b) / (t_1 / y)) + a);
	} else {
		tmp = ((t_2 + ((z * y) + (z * x))) - (b * y)) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = y + (t + x)
    t_2 = a * (y + t)
    t_3 = ((t_2 + (z * (y + x))) - (b * y)) / t_1
    if ((t_3 <= (-5d+124)) .or. (.not. (t_3 <= 1d+249))) then
        tmp = (z / (t_1 / x)) + (((z - b) / (t_1 / y)) + a)
    else
        tmp = ((t_2 + ((z * y) + (z * x))) - (b * y)) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (t + x);
	double t_2 = a * (y + t);
	double t_3 = ((t_2 + (z * (y + x))) - (b * y)) / t_1;
	double tmp;
	if ((t_3 <= -5e+124) || !(t_3 <= 1e+249)) {
		tmp = (z / (t_1 / x)) + (((z - b) / (t_1 / y)) + a);
	} else {
		tmp = ((t_2 + ((z * y) + (z * x))) - (b * y)) / t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = y + (t + x)
	t_2 = a * (y + t)
	t_3 = ((t_2 + (z * (y + x))) - (b * y)) / t_1
	tmp = 0
	if (t_3 <= -5e+124) or not (t_3 <= 1e+249):
		tmp = (z / (t_1 / x)) + (((z - b) / (t_1 / y)) + a)
	else:
		tmp = ((t_2 + ((z * y) + (z * x))) - (b * y)) / t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(t + x))
	t_2 = Float64(a * Float64(y + t))
	t_3 = Float64(Float64(Float64(t_2 + Float64(z * Float64(y + x))) - Float64(b * y)) / t_1)
	tmp = 0.0
	if ((t_3 <= -5e+124) || !(t_3 <= 1e+249))
		tmp = Float64(Float64(z / Float64(t_1 / x)) + Float64(Float64(Float64(z - b) / Float64(t_1 / y)) + a));
	else
		tmp = Float64(Float64(Float64(t_2 + Float64(Float64(z * y) + Float64(z * x))) - Float64(b * y)) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = y + (t + x);
	t_2 = a * (y + t);
	t_3 = ((t_2 + (z * (y + x))) - (b * y)) / t_1;
	tmp = 0.0;
	if ((t_3 <= -5e+124) || ~((t_3 <= 1e+249)))
		tmp = (z / (t_1 / x)) + (((z - b) / (t_1 / y)) + a);
	else
		tmp = ((t_2 + ((z * y) + (z * x))) - (b * y)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(t + x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(t$95$2 + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$3, -5e+124], N[Not[LessEqual[t$95$3, 1e+249]], $MachinePrecision]], N[(N[(z / N[(t$95$1 / x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(z - b), $MachinePrecision] / N[(t$95$1 / y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$2 + N[(N[(z * y), $MachinePrecision] + N[(z * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -4.9999999999999996e124 or 9.9999999999999992e248 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 24.6%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Step-by-step derivation

   3. Simplified 25.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, z, \mathsf{fma}\left(y, a - \left(b - z\right), t \cdot a\right)\right)}{x + \left(y + t\right)}} \]

   4. Taylor expanded in a around inf 24.5%

      \[\leadsto \color{blue}{\frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \left(\frac{z \cdot x}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 24.5%

         \[\leadsto \frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \color{blue}{\left(\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)} + \frac{z \cdot x}{y + \left(t + x\right)}\right)} \]

      2. associate-+r+ 24.5%

         \[\leadsto \color{blue}{\left(\frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right) + \frac{z \cdot x}{y + \left(t + x\right)}} \]

      3. associate-/l* 44.3%

         \[\leadsto \left(\color{blue}{\frac{z - b}{\frac{y + \left(t + x\right)}{y}}} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right) + \frac{z \cdot x}{y + \left(t + x\right)} \]

      4. associate-/l* 71.8%

         \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \color{blue}{\frac{a}{\frac{y + \left(t + x\right)}{y + t}}}\right) + \frac{z \cdot x}{y + \left(t + x\right)} \]

      5. associate-/l* 99.9%

         \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{a}{\frac{y + \left(t + x\right)}{y + t}}\right) + \color{blue}{\frac{z}{\frac{y + \left(t + x\right)}{x}}} \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{a}{\frac{y + \left(t + x\right)}{y + t}}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}}} \]

   7. Taylor expanded in y around inf 89.5%

      \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \color{blue}{a}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}} \]

   if -4.9999999999999996e124 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 9.9999999999999992e248

   1. Initial program 98.9%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in x around 0 99.0%

      \[\leadsto \frac{\left(\color{blue}{\left(y \cdot z + z \cdot x\right)} + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(y + x\right)\right) - b \cdot y}{y + \left(t + x\right)} \leq -5 \cdot 10^{+124} \lor \neg \left(\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(y + x\right)\right) - b \cdot y}{y + \left(t + x\right)} \leq 10^{+249}\right):\\ \;\;\;\;\frac{z}{\frac{y + \left(t + x\right)}{x}} + \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(a \cdot \left(y + t\right) + \left(z \cdot y + z \cdot x\right)\right) - b \cdot y}{y + \left(t + x\right)}\\ \end{array} \]

Alternative 5: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(a \cdot \left(y + t\right) + z \cdot \left(y + x\right)\right) - b \cdot y}{y + \left(t + x\right)}\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+247}:\\ \;\;\;\;a + \frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+257}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- (+ (* a (+ y t)) (* z (+ y x))) (* b y)) (+ y (+ t x)))))
   (if (<= t_1 -5e+247)
     (+ a (/ (- z b) (/ (+ y t) y)))
     (if (<= t_1 2e+257) t_1 (- (+ z a) b)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((a * (y + t)) + (z * (y + x))) - (b * y)) / (y + (t + x));
	double tmp;
	if (t_1 <= -5e+247) {
		tmp = a + ((z - b) / ((y + t) / y));
	} else if (t_1 <= 2e+257) {
		tmp = t_1;
	} else {
		tmp = (z + a) - b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (((a * (y + t)) + (z * (y + x))) - (b * y)) / (y + (t + x))
    if (t_1 <= (-5d+247)) then
        tmp = a + ((z - b) / ((y + t) / y))
    else if (t_1 <= 2d+257) then
        tmp = t_1
    else
        tmp = (z + a) - b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (((a * (y + t)) + (z * (y + x))) - (b * y)) / (y + (t + x));
	double tmp;
	if (t_1 <= -5e+247) {
		tmp = a + ((z - b) / ((y + t) / y));
	} else if (t_1 <= 2e+257) {
		tmp = t_1;
	} else {
		tmp = (z + a) - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (((a * (y + t)) + (z * (y + x))) - (b * y)) / (y + (t + x))
	tmp = 0
	if t_1 <= -5e+247:
		tmp = a + ((z - b) / ((y + t) / y))
	elif t_1 <= 2e+257:
		tmp = t_1
	else:
		tmp = (z + a) - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(Float64(Float64(a * Float64(y + t)) + Float64(z * Float64(y + x))) - Float64(b * y)) / Float64(y + Float64(t + x)))
	tmp = 0.0
	if (t_1 <= -5e+247)
		tmp = Float64(a + Float64(Float64(z - b) / Float64(Float64(y + t) / y)));
	elseif (t_1 <= 2e+257)
		tmp = t_1;
	else
		tmp = Float64(Float64(z + a) - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (((a * (y + t)) + (z * (y + x))) - (b * y)) / (y + (t + x));
	tmp = 0.0;
	if (t_1 <= -5e+247)
		tmp = a + ((z - b) / ((y + t) / y));
	elseif (t_1 <= 2e+257)
		tmp = t_1;
	else
		tmp = (z + a) - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(N[(N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * y), $MachinePrecision]), $MachinePrecision] / N[(y + N[(t + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+247], N[(a + N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+257], t$95$1, N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < -5.00000000000000023e247

   1. Initial program 17.6%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Step-by-step derivation

   3. Simplified 17.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, z, \mathsf{fma}\left(y, a - \left(b - z\right), t \cdot a\right)\right)}{x + \left(y + t\right)}} \]

   4. Taylor expanded in x around 0 20.0%

      \[\leadsto \color{blue}{\frac{a \cdot t + y \cdot \left(\left(a + z\right) - b\right)}{y + t}} \]

   5. Taylor expanded in a around 0 40.7%

      \[\leadsto \color{blue}{a + \frac{\left(z - b\right) \cdot y}{y + t}} \]
   6. Step-by-step derivation

      1. +-commutative 40.7%

         \[\leadsto \color{blue}{\frac{\left(z - b\right) \cdot y}{y + t} + a} \]

      2. associate-/l* 75.4%

         \[\leadsto \color{blue}{\frac{z - b}{\frac{y + t}{y}}} + a \]

   7. Simplified 75.4%

      \[\leadsto \color{blue}{\frac{z - b}{\frac{y + t}{y}} + a} \]

   if -5.00000000000000023e247 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y)) < 2.00000000000000006e257

   1. Initial program 99.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   if 2.00000000000000006e257 < (/.f64 (-.f64 (+.f64 (*.f64 (+.f64 x y) z) (*.f64 (+.f64 t y) a)) (*.f64 y b)) (+.f64 (+.f64 x t) y))

   1. Initial program 7.6%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in y around inf 74.9%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]
   3. Step-by-step derivation

      1. +-commutative 74.9%

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   4. Simplified 74.9%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(y + x\right)\right) - b \cdot y}{y + \left(t + x\right)} \leq -5 \cdot 10^{+247}:\\ \;\;\;\;a + \frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(y + x\right)\right) - b \cdot y}{y + \left(t + x\right)} \leq 2 \cdot 10^{+257}:\\ \;\;\;\;\frac{\left(a \cdot \left(y + t\right) + z \cdot \left(y + x\right)\right) - b \cdot y}{y + \left(t + x\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Alternative 6: 74.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + \frac{z - b}{\frac{y + t}{y}}\\ t_2 := y + \left(t + x\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{+64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-110}:\\ \;\;\;\;\left(\frac{z - b}{\frac{t_2}{y}} + a\right) + \frac{z \cdot x}{y + x}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-130} \lor \neg \left(y \leq 2.6 \cdot 10^{-78}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(y + t\right) + z \cdot \left(y + x\right)}{t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ a (/ (- z b) (/ (+ y t) y)))) (t_2 (+ y (+ t x))))
   (if (<= y -4e+64)
     t_1
     (if (<= y -2.4e-110)
       (+ (+ (/ (- z b) (/ t_2 y)) a) (/ (* z x) (+ y x)))
       (if (or (<= y -1.4e-130) (not (<= y 2.6e-78)))
         t_1
         (/ (+ (* a (+ y t)) (* z (+ y x))) t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a + ((z - b) / ((y + t) / y));
	double t_2 = y + (t + x);
	double tmp;
	if (y <= -4e+64) {
		tmp = t_1;
	} else if (y <= -2.4e-110) {
		tmp = (((z - b) / (t_2 / y)) + a) + ((z * x) / (y + x));
	} else if ((y <= -1.4e-130) || !(y <= 2.6e-78)) {
		tmp = t_1;
	} else {
		tmp = ((a * (y + t)) + (z * (y + x))) / t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a + ((z - b) / ((y + t) / y))
    t_2 = y + (t + x)
    if (y <= (-4d+64)) then
        tmp = t_1
    else if (y <= (-2.4d-110)) then
        tmp = (((z - b) / (t_2 / y)) + a) + ((z * x) / (y + x))
    else if ((y <= (-1.4d-130)) .or. (.not. (y <= 2.6d-78))) then
        tmp = t_1
    else
        tmp = ((a * (y + t)) + (z * (y + x))) / t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a + ((z - b) / ((y + t) / y));
	double t_2 = y + (t + x);
	double tmp;
	if (y <= -4e+64) {
		tmp = t_1;
	} else if (y <= -2.4e-110) {
		tmp = (((z - b) / (t_2 / y)) + a) + ((z * x) / (y + x));
	} else if ((y <= -1.4e-130) || !(y <= 2.6e-78)) {
		tmp = t_1;
	} else {
		tmp = ((a * (y + t)) + (z * (y + x))) / t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a + ((z - b) / ((y + t) / y))
	t_2 = y + (t + x)
	tmp = 0
	if y <= -4e+64:
		tmp = t_1
	elif y <= -2.4e-110:
		tmp = (((z - b) / (t_2 / y)) + a) + ((z * x) / (y + x))
	elif (y <= -1.4e-130) or not (y <= 2.6e-78):
		tmp = t_1
	else:
		tmp = ((a * (y + t)) + (z * (y + x))) / t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a + Float64(Float64(z - b) / Float64(Float64(y + t) / y)))
	t_2 = Float64(y + Float64(t + x))
	tmp = 0.0
	if (y <= -4e+64)
		tmp = t_1;
	elseif (y <= -2.4e-110)
		tmp = Float64(Float64(Float64(Float64(z - b) / Float64(t_2 / y)) + a) + Float64(Float64(z * x) / Float64(y + x)));
	elseif ((y <= -1.4e-130) || !(y <= 2.6e-78))
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(a * Float64(y + t)) + Float64(z * Float64(y + x))) / t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a + ((z - b) / ((y + t) / y));
	t_2 = y + (t + x);
	tmp = 0.0;
	if (y <= -4e+64)
		tmp = t_1;
	elseif (y <= -2.4e-110)
		tmp = (((z - b) / (t_2 / y)) + a) + ((z * x) / (y + x));
	elseif ((y <= -1.4e-130) || ~((y <= 2.6e-78)))
		tmp = t_1;
	else
		tmp = ((a * (y + t)) + (z * (y + x))) / t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a + N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y + N[(t + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+64], t$95$1, If[LessEqual[y, -2.4e-110], N[(N[(N[(N[(z - b), $MachinePrecision] / N[(t$95$2 / y), $MachinePrecision]), $MachinePrecision] + a), $MachinePrecision] + N[(N[(z * x), $MachinePrecision] / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -1.4e-130], N[Not[LessEqual[y, 2.6e-78]], $MachinePrecision]], t$95$1, N[(N[(N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.00000000000000009e64 or -2.40000000000000006e-110 < y < -1.40000000000000008e-130 or 2.6000000000000001e-78 < y

   1. Initial program 38.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Step-by-step derivation

   3. Simplified 38.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, z, \mathsf{fma}\left(y, a - \left(b - z\right), t \cdot a\right)\right)}{x + \left(y + t\right)}} \]

   4. Taylor expanded in x around 0 34.7%

      \[\leadsto \color{blue}{\frac{a \cdot t + y \cdot \left(\left(a + z\right) - b\right)}{y + t}} \]

   5. Taylor expanded in a around 0 46.4%

      \[\leadsto \color{blue}{a + \frac{\left(z - b\right) \cdot y}{y + t}} \]
   6. Step-by-step derivation

      1. +-commutative 46.4%

         \[\leadsto \color{blue}{\frac{\left(z - b\right) \cdot y}{y + t} + a} \]

      2. associate-/l* 84.8%

         \[\leadsto \color{blue}{\frac{z - b}{\frac{y + t}{y}}} + a \]

   7. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{z - b}{\frac{y + t}{y}} + a} \]

   if -4.00000000000000009e64 < y < -2.40000000000000006e-110

   1. Initial program 73.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Step-by-step derivation

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, z, \mathsf{fma}\left(y, a - \left(b - z\right), t \cdot a\right)\right)}{x + \left(y + t\right)}} \]

   4. Taylor expanded in a around inf 72.8%

      \[\leadsto \color{blue}{\frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \left(\frac{z \cdot x}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right)} \]
   5. Step-by-step derivation

      1. +-commutative 72.8%

         \[\leadsto \frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \color{blue}{\left(\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)} + \frac{z \cdot x}{y + \left(t + x\right)}\right)} \]

      2. associate-+r+ 72.8%

         \[\leadsto \color{blue}{\left(\frac{\left(z - b\right) \cdot y}{y + \left(t + x\right)} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right) + \frac{z \cdot x}{y + \left(t + x\right)}} \]

      3. associate-/l* 73.2%

         \[\leadsto \left(\color{blue}{\frac{z - b}{\frac{y + \left(t + x\right)}{y}}} + \frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}\right) + \frac{z \cdot x}{y + \left(t + x\right)} \]

      4. associate-/l* 84.6%

         \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \color{blue}{\frac{a}{\frac{y + \left(t + x\right)}{y + t}}}\right) + \frac{z \cdot x}{y + \left(t + x\right)} \]

      5. associate-/l* 99.9%

         \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{a}{\frac{y + \left(t + x\right)}{y + t}}\right) + \color{blue}{\frac{z}{\frac{y + \left(t + x\right)}{x}}} \]

   6. Simplified 99.9%

      \[\leadsto \color{blue}{\left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \frac{a}{\frac{y + \left(t + x\right)}{y + t}}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}}} \]

   7. Taylor expanded in y around inf 83.7%

      \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + \color{blue}{a}\right) + \frac{z}{\frac{y + \left(t + x\right)}{x}} \]

   8. Taylor expanded in t around 0 66.5%

      \[\leadsto \left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + a\right) + \color{blue}{\frac{z \cdot x}{y + x}} \]

   if -1.40000000000000008e-130 < y < 2.6000000000000001e-78

   1. Initial program 81.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in b around 0 74.0%

      \[\leadsto \frac{\color{blue}{a \cdot \left(y + t\right) + \left(y + x\right) \cdot z}}{\left(x + t\right) + y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+64}:\\ \;\;\;\;a + \frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-110}:\\ \;\;\;\;\left(\frac{z - b}{\frac{y + \left(t + x\right)}{y}} + a\right) + \frac{z \cdot x}{y + x}\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-130} \lor \neg \left(y \leq 2.6 \cdot 10^{-78}\right):\\ \;\;\;\;a + \frac{z - b}{\frac{y + t}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(y + t\right) + z \cdot \left(y + x\right)}{y + \left(t + x\right)}\\ \end{array} \]

Alternative 7: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-75} \lor \neg \left(y \leq 8 \cdot 10^{-79}\right):\\ \;\;\;\;a + \frac{z - b}{\frac{y + t}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(y + t\right) + z \cdot \left(y + x\right)}{y + \left(t + x\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -1.85e-75) (not (<= y 8e-79)))
   (+ a (/ (- z b) (/ (+ y t) y)))
   (/ (+ (* a (+ y t)) (* z (+ y x))) (+ y (+ t x)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.85e-75) || !(y <= 8e-79)) {
		tmp = a + ((z - b) / ((y + t) / y));
	} else {
		tmp = ((a * (y + t)) + (z * (y + x))) / (y + (t + x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-1.85d-75)) .or. (.not. (y <= 8d-79))) then
        tmp = a + ((z - b) / ((y + t) / y))
    else
        tmp = ((a * (y + t)) + (z * (y + x))) / (y + (t + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -1.85e-75) || !(y <= 8e-79)) {
		tmp = a + ((z - b) / ((y + t) / y));
	} else {
		tmp = ((a * (y + t)) + (z * (y + x))) / (y + (t + x));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -1.85e-75) or not (y <= 8e-79):
		tmp = a + ((z - b) / ((y + t) / y))
	else:
		tmp = ((a * (y + t)) + (z * (y + x))) / (y + (t + x))
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -1.85e-75) || !(y <= 8e-79))
		tmp = Float64(a + Float64(Float64(z - b) / Float64(Float64(y + t) / y)));
	else
		tmp = Float64(Float64(Float64(a * Float64(y + t)) + Float64(z * Float64(y + x))) / Float64(y + Float64(t + x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -1.85e-75) || ~((y <= 8e-79)))
		tmp = a + ((z - b) / ((y + t) / y));
	else
		tmp = ((a * (y + t)) + (z * (y + x))) / (y + (t + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -1.85e-75], N[Not[LessEqual[y, 8e-79]], $MachinePrecision]], N[(a + N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(a * N[(y + t), $MachinePrecision]), $MachinePrecision] + N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(t + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.85000000000000012e-75 or 8e-79 < y

   1. Initial program 43.6%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Step-by-step derivation

   3. Simplified 44.1%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, z, \mathsf{fma}\left(y, a - \left(b - z\right), t \cdot a\right)\right)}{x + \left(y + t\right)}} \]

   4. Taylor expanded in x around 0 36.7%

      \[\leadsto \color{blue}{\frac{a \cdot t + y \cdot \left(\left(a + z\right) - b\right)}{y + t}} \]

   5. Taylor expanded in a around 0 47.6%

      \[\leadsto \color{blue}{a + \frac{\left(z - b\right) \cdot y}{y + t}} \]
   6. Step-by-step derivation

      1. +-commutative 47.6%

         \[\leadsto \color{blue}{\frac{\left(z - b\right) \cdot y}{y + t} + a} \]

      2. associate-/l* 79.2%

         \[\leadsto \color{blue}{\frac{z - b}{\frac{y + t}{y}}} + a \]

   7. Simplified 79.2%

      \[\leadsto \color{blue}{\frac{z - b}{\frac{y + t}{y}} + a} \]

   if -1.85000000000000012e-75 < y < 8e-79

   1. Initial program 80.8%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in b around 0 72.0%

      \[\leadsto \frac{\color{blue}{a \cdot \left(y + t\right) + \left(y + x\right) \cdot z}}{\left(x + t\right) + y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-75} \lor \neg \left(y \leq 8 \cdot 10^{-79}\right):\\ \;\;\;\;a + \frac{z - b}{\frac{y + t}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{a \cdot \left(y + t\right) + z \cdot \left(y + x\right)}{y + \left(t + x\right)}\\ \end{array} \]

Alternative 8: 73.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a + \frac{z - b}{\frac{y + t}{y}}\\ t_2 := z + t \cdot \left(\frac{a}{x} - \frac{z}{x}\right)\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{+86}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+196}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+232}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ a (/ (- z b) (/ (+ y t) y))))
        (t_2 (+ z (* t (- (/ a x) (/ z x))))))
   (if (<= x -4.1e+86)
     t_2
     (if (<= x 6.8e+92)
       t_1
       (if (<= x 8.2e+196) (- (+ z a) b) (if (<= x 6.6e+232) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a + ((z - b) / ((y + t) / y));
	double t_2 = z + (t * ((a / x) - (z / x)));
	double tmp;
	if (x <= -4.1e+86) {
		tmp = t_2;
	} else if (x <= 6.8e+92) {
		tmp = t_1;
	} else if (x <= 8.2e+196) {
		tmp = (z + a) - b;
	} else if (x <= 6.6e+232) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a + ((z - b) / ((y + t) / y))
    t_2 = z + (t * ((a / x) - (z / x)))
    if (x <= (-4.1d+86)) then
        tmp = t_2
    else if (x <= 6.8d+92) then
        tmp = t_1
    else if (x <= 8.2d+196) then
        tmp = (z + a) - b
    else if (x <= 6.6d+232) then
        tmp = t_1
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = a + ((z - b) / ((y + t) / y));
	double t_2 = z + (t * ((a / x) - (z / x)));
	double tmp;
	if (x <= -4.1e+86) {
		tmp = t_2;
	} else if (x <= 6.8e+92) {
		tmp = t_1;
	} else if (x <= 8.2e+196) {
		tmp = (z + a) - b;
	} else if (x <= 6.6e+232) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = a + ((z - b) / ((y + t) / y))
	t_2 = z + (t * ((a / x) - (z / x)))
	tmp = 0
	if x <= -4.1e+86:
		tmp = t_2
	elif x <= 6.8e+92:
		tmp = t_1
	elif x <= 8.2e+196:
		tmp = (z + a) - b
	elif x <= 6.6e+232:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(a + Float64(Float64(z - b) / Float64(Float64(y + t) / y)))
	t_2 = Float64(z + Float64(t * Float64(Float64(a / x) - Float64(z / x))))
	tmp = 0.0
	if (x <= -4.1e+86)
		tmp = t_2;
	elseif (x <= 6.8e+92)
		tmp = t_1;
	elseif (x <= 8.2e+196)
		tmp = Float64(Float64(z + a) - b);
	elseif (x <= 6.6e+232)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = a + ((z - b) / ((y + t) / y));
	t_2 = z + (t * ((a / x) - (z / x)));
	tmp = 0.0;
	if (x <= -4.1e+86)
		tmp = t_2;
	elseif (x <= 6.8e+92)
		tmp = t_1;
	elseif (x <= 8.2e+196)
		tmp = (z + a) - b;
	elseif (x <= 6.6e+232)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(a + N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z + N[(t * N[(N[(a / x), $MachinePrecision] - N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.1e+86], t$95$2, If[LessEqual[x, 6.8e+92], t$95$1, If[LessEqual[x, 8.2e+196], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision], If[LessEqual[x, 6.6e+232], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.0999999999999999e86 or 6.6e232 < x

   1. Initial program 44.8%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in y around 0 39.7%

      \[\leadsto \color{blue}{\frac{z \cdot x + a \cdot t}{t + x}} \]

   3. Taylor expanded in t around 0 69.9%

      \[\leadsto \color{blue}{\left(\frac{a}{x} - \frac{z}{x}\right) \cdot t + z} \]

   if -4.0999999999999999e86 < x < 6.7999999999999996e92 or 8.1999999999999993e196 < x < 6.6e232

   1. Initial program 64.2%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Step-by-step derivation

   3. Simplified 64.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, z, \mathsf{fma}\left(y, a - \left(b - z\right), t \cdot a\right)\right)}{x + \left(y + t\right)}} \]

   4. Taylor expanded in x around 0 50.2%

      \[\leadsto \color{blue}{\frac{a \cdot t + y \cdot \left(\left(a + z\right) - b\right)}{y + t}} \]

   5. Taylor expanded in a around 0 60.5%

      \[\leadsto \color{blue}{a + \frac{\left(z - b\right) \cdot y}{y + t}} \]
   6. Step-by-step derivation

      1. +-commutative 60.5%

         \[\leadsto \color{blue}{\frac{\left(z - b\right) \cdot y}{y + t} + a} \]

      2. associate-/l* 81.0%

         \[\leadsto \color{blue}{\frac{z - b}{\frac{y + t}{y}}} + a \]

   7. Simplified 81.0%

      \[\leadsto \color{blue}{\frac{z - b}{\frac{y + t}{y}} + a} \]

   if 6.7999999999999996e92 < x < 8.1999999999999993e196

   1. Initial program 34.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in y around inf 70.8%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]
   3. Step-by-step derivation

      1. +-commutative 70.8%

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   4. Simplified 70.8%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+86}:\\ \;\;\;\;z + t \cdot \left(\frac{a}{x} - \frac{z}{x}\right)\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{+92}:\\ \;\;\;\;a + \frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+196}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+232}:\\ \;\;\;\;a + \frac{z - b}{\frac{y + t}{y}}\\ \mathbf{else}:\\ \;\;\;\;z + t \cdot \left(\frac{a}{x} - \frac{z}{x}\right)\\ \end{array} \]

Alternative 9: 73.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z + t \cdot \left(\frac{a}{x} - \frac{z}{x}\right)\\ \mathbf{if}\;x \leq -4.1 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+94}:\\ \;\;\;\;a + \frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+200}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+228}:\\ \;\;\;\;\frac{a}{\frac{y + \left(t + x\right)}{y + t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ z (* t (- (/ a x) (/ z x))))))
   (if (<= x -4.1e+86)
     t_1
     (if (<= x 1.2e+94)
       (+ a (/ (- z b) (/ (+ y t) y)))
       (if (<= x 1.42e+200)
         (- (+ z a) b)
         (if (<= x 2.9e+228) (/ a (/ (+ y (+ t x)) (+ y t))) t_1))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z + (t * ((a / x) - (z / x)));
	double tmp;
	if (x <= -4.1e+86) {
		tmp = t_1;
	} else if (x <= 1.2e+94) {
		tmp = a + ((z - b) / ((y + t) / y));
	} else if (x <= 1.42e+200) {
		tmp = (z + a) - b;
	} else if (x <= 2.9e+228) {
		tmp = a / ((y + (t + x)) / (y + t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z + (t * ((a / x) - (z / x)))
    if (x <= (-4.1d+86)) then
        tmp = t_1
    else if (x <= 1.2d+94) then
        tmp = a + ((z - b) / ((y + t) / y))
    else if (x <= 1.42d+200) then
        tmp = (z + a) - b
    else if (x <= 2.9d+228) then
        tmp = a / ((y + (t + x)) / (y + t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = z + (t * ((a / x) - (z / x)));
	double tmp;
	if (x <= -4.1e+86) {
		tmp = t_1;
	} else if (x <= 1.2e+94) {
		tmp = a + ((z - b) / ((y + t) / y));
	} else if (x <= 1.42e+200) {
		tmp = (z + a) - b;
	} else if (x <= 2.9e+228) {
		tmp = a / ((y + (t + x)) / (y + t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = z + (t * ((a / x) - (z / x)))
	tmp = 0
	if x <= -4.1e+86:
		tmp = t_1
	elif x <= 1.2e+94:
		tmp = a + ((z - b) / ((y + t) / y))
	elif x <= 1.42e+200:
		tmp = (z + a) - b
	elif x <= 2.9e+228:
		tmp = a / ((y + (t + x)) / (y + t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(z + Float64(t * Float64(Float64(a / x) - Float64(z / x))))
	tmp = 0.0
	if (x <= -4.1e+86)
		tmp = t_1;
	elseif (x <= 1.2e+94)
		tmp = Float64(a + Float64(Float64(z - b) / Float64(Float64(y + t) / y)));
	elseif (x <= 1.42e+200)
		tmp = Float64(Float64(z + a) - b);
	elseif (x <= 2.9e+228)
		tmp = Float64(a / Float64(Float64(y + Float64(t + x)) / Float64(y + t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = z + (t * ((a / x) - (z / x)));
	tmp = 0.0;
	if (x <= -4.1e+86)
		tmp = t_1;
	elseif (x <= 1.2e+94)
		tmp = a + ((z - b) / ((y + t) / y));
	elseif (x <= 1.42e+200)
		tmp = (z + a) - b;
	elseif (x <= 2.9e+228)
		tmp = a / ((y + (t + x)) / (y + t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(z + N[(t * N[(N[(a / x), $MachinePrecision] - N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.1e+86], t$95$1, If[LessEqual[x, 1.2e+94], N[(a + N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.42e+200], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision], If[LessEqual[x, 2.9e+228], N[(a / N[(N[(y + N[(t + x), $MachinePrecision]), $MachinePrecision] / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.0999999999999999e86 or 2.90000000000000002e228 < x

   1. Initial program 43.4%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in y around 0 38.6%

      \[\leadsto \color{blue}{\frac{z \cdot x + a \cdot t}{t + x}} \]

   3. Taylor expanded in t around 0 69.3%

      \[\leadsto \color{blue}{\left(\frac{a}{x} - \frac{z}{x}\right) \cdot t + z} \]

   if -4.0999999999999999e86 < x < 1.19999999999999991e94

   1. Initial program 66.6%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Step-by-step derivation

   3. Simplified 66.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, z, \mathsf{fma}\left(y, a - \left(b - z\right), t \cdot a\right)\right)}{x + \left(y + t\right)}} \]

   4. Taylor expanded in x around 0 52.3%

      \[\leadsto \color{blue}{\frac{a \cdot t + y \cdot \left(\left(a + z\right) - b\right)}{y + t}} \]

   5. Taylor expanded in a around 0 61.6%

      \[\leadsto \color{blue}{a + \frac{\left(z - b\right) \cdot y}{y + t}} \]
   6. Step-by-step derivation

      1. +-commutative 61.6%

         \[\leadsto \color{blue}{\frac{\left(z - b\right) \cdot y}{y + t} + a} \]

      2. associate-/l* 81.6%

         \[\leadsto \color{blue}{\frac{z - b}{\frac{y + t}{y}}} + a \]

   7. Simplified 81.6%

      \[\leadsto \color{blue}{\frac{z - b}{\frac{y + t}{y}} + a} \]

   if 1.19999999999999991e94 < x < 1.42e200

   1. Initial program 34.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in y around inf 70.8%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]
   3. Step-by-step derivation

      1. +-commutative 70.8%

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   4. Simplified 70.8%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]

   if 1.42e200 < x < 2.90000000000000002e228

   1. Initial program 27.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in a around inf 16.7%

      \[\leadsto \color{blue}{\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}} \]
   3. Step-by-step derivation

      1. associate-/l* 75.9%

         \[\leadsto \color{blue}{\frac{a}{\frac{y + \left(t + x\right)}{y + t}}} \]

   4. Simplified 75.9%

      \[\leadsto \color{blue}{\frac{a}{\frac{y + \left(t + x\right)}{y + t}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.1 \cdot 10^{+86}:\\ \;\;\;\;z + t \cdot \left(\frac{a}{x} - \frac{z}{x}\right)\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+94}:\\ \;\;\;\;a + \frac{z - b}{\frac{y + t}{y}}\\ \mathbf{elif}\;x \leq 1.42 \cdot 10^{+200}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{elif}\;x \leq 2.9 \cdot 10^{+228}:\\ \;\;\;\;\frac{a}{\frac{y + \left(t + x\right)}{y + t}}\\ \mathbf{else}:\\ \;\;\;\;z + t \cdot \left(\frac{a}{x} - \frac{z}{x}\right)\\ \end{array} \]

Alternative 10: 73.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-100} \lor \neg \left(y \leq 2.9 \cdot 10^{-135}\right):\\ \;\;\;\;a + \frac{z - b}{\frac{y + t}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x + t \cdot a}{t + x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= y -6.6e-100) (not (<= y 2.9e-135)))
   (+ a (/ (- z b) (/ (+ y t) y)))
   (/ (+ (* z x) (* t a)) (+ t x))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.6e-100) || !(y <= 2.9e-135)) {
		tmp = a + ((z - b) / ((y + t) / y));
	} else {
		tmp = ((z * x) + (t * a)) / (t + x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((y <= (-6.6d-100)) .or. (.not. (y <= 2.9d-135))) then
        tmp = a + ((z - b) / ((y + t) / y))
    else
        tmp = ((z * x) + (t * a)) / (t + x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((y <= -6.6e-100) || !(y <= 2.9e-135)) {
		tmp = a + ((z - b) / ((y + t) / y));
	} else {
		tmp = ((z * x) + (t * a)) / (t + x);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (y <= -6.6e-100) or not (y <= 2.9e-135):
		tmp = a + ((z - b) / ((y + t) / y))
	else:
		tmp = ((z * x) + (t * a)) / (t + x)
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((y <= -6.6e-100) || !(y <= 2.9e-135))
		tmp = Float64(a + Float64(Float64(z - b) / Float64(Float64(y + t) / y)));
	else
		tmp = Float64(Float64(Float64(z * x) + Float64(t * a)) / Float64(t + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((y <= -6.6e-100) || ~((y <= 2.9e-135)))
		tmp = a + ((z - b) / ((y + t) / y));
	else
		tmp = ((z * x) + (t * a)) / (t + x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[y, -6.6e-100], N[Not[LessEqual[y, 2.9e-135]], $MachinePrecision]], N[(a + N[(N[(z - b), $MachinePrecision] / N[(N[(y + t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(z * x), $MachinePrecision] + N[(t * a), $MachinePrecision]), $MachinePrecision] / N[(t + x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.59999999999999993e-100 or 2.9000000000000002e-135 < y

   1. Initial program 48.2%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]
   2. Step-by-step derivation

   3. Simplified 48.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, z, \mathsf{fma}\left(y, a - \left(b - z\right), t \cdot a\right)\right)}{x + \left(y + t\right)}} \]

   4. Taylor expanded in x around 0 38.8%

      \[\leadsto \color{blue}{\frac{a \cdot t + y \cdot \left(\left(a + z\right) - b\right)}{y + t}} \]

   5. Taylor expanded in a around 0 49.5%

      \[\leadsto \color{blue}{a + \frac{\left(z - b\right) \cdot y}{y + t}} \]
   6. Step-by-step derivation

      1. +-commutative 49.5%

         \[\leadsto \color{blue}{\frac{\left(z - b\right) \cdot y}{y + t} + a} \]

      2. associate-/l* 77.3%

         \[\leadsto \color{blue}{\frac{z - b}{\frac{y + t}{y}}} + a \]

   7. Simplified 77.3%

      \[\leadsto \color{blue}{\frac{z - b}{\frac{y + t}{y}} + a} \]

   if -6.59999999999999993e-100 < y < 2.9000000000000002e-135

   1. Initial program 80.5%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in y around 0 69.0%

      \[\leadsto \color{blue}{\frac{z \cdot x + a \cdot t}{t + x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-100} \lor \neg \left(y \leq 2.9 \cdot 10^{-135}\right):\\ \;\;\;\;a + \frac{z - b}{\frac{y + t}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x + t \cdot a}{t + x}\\ \end{array} \]

Alternative 11: 60.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+87} \lor \neg \left(t \leq 2.2 \cdot 10^{+149}\right):\\ \;\;\;\;\frac{a}{1 + \frac{x}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (or (<= t -2.6e+87) (not (<= t 2.2e+149)))
   (/ a (+ 1.0 (/ x t)))
   (- (+ z a) b)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -2.6e+87) || !(t <= 2.2e+149)) {
		tmp = a / (1.0 + (x / t));
	} else {
		tmp = (z + a) - b;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if ((t <= (-2.6d+87)) .or. (.not. (t <= 2.2d+149))) then
        tmp = a / (1.0d0 + (x / t))
    else
        tmp = (z + a) - b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if ((t <= -2.6e+87) || !(t <= 2.2e+149)) {
		tmp = a / (1.0 + (x / t));
	} else {
		tmp = (z + a) - b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if (t <= -2.6e+87) or not (t <= 2.2e+149):
		tmp = a / (1.0 + (x / t))
	else:
		tmp = (z + a) - b
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if ((t <= -2.6e+87) || !(t <= 2.2e+149))
		tmp = Float64(a / Float64(1.0 + Float64(x / t)));
	else
		tmp = Float64(Float64(z + a) - b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if ((t <= -2.6e+87) || ~((t <= 2.2e+149)))
		tmp = a / (1.0 + (x / t));
	else
		tmp = (z + a) - b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[Or[LessEqual[t, -2.6e+87], N[Not[LessEqual[t, 2.2e+149]], $MachinePrecision]], N[(a / N[(1.0 + N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.59999999999999998e87 or 2.2e149 < t

   1. Initial program 49.2%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in a around inf 32.8%

      \[\leadsto \color{blue}{\frac{a \cdot \left(y + t\right)}{y + \left(t + x\right)}} \]
   3. Step-by-step derivation

      1. associate-/l* 64.3%

         \[\leadsto \color{blue}{\frac{a}{\frac{y + \left(t + x\right)}{y + t}}} \]

   4. Simplified 64.3%

      \[\leadsto \color{blue}{\frac{a}{\frac{y + \left(t + x\right)}{y + t}}} \]

   5. Taylor expanded in t around inf 63.0%

      \[\leadsto \frac{a}{\color{blue}{1 + \frac{x}{t}}} \]

   if -2.59999999999999998e87 < t < 2.2e149

   1. Initial program 61.7%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in y around inf 64.4%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]
   3. Step-by-step derivation

      1. +-commutative 64.4%

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   4. Simplified 64.4%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{+87} \lor \neg \left(t \leq 2.2 \cdot 10^{+149}\right):\\ \;\;\;\;\frac{a}{1 + \frac{x}{t}}\\ \mathbf{else}:\\ \;\;\;\;\left(z + a\right) - b\\ \end{array} \]

Alternative 12: 60.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{+164}:\\ \;\;\;\;a\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+161}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= t -3.5e+164) a (if (<= t 9.5e+161) (- (+ z a) b) a)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -3.5e+164) {
		tmp = a;
	} else if (t <= 9.5e+161) {
		tmp = (z + a) - b;
	} else {
		tmp = a;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (t <= (-3.5d+164)) then
        tmp = a
    else if (t <= 9.5d+161) then
        tmp = (z + a) - b
    else
        tmp = a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (t <= -3.5e+164) {
		tmp = a;
	} else if (t <= 9.5e+161) {
		tmp = (z + a) - b;
	} else {
		tmp = a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if t <= -3.5e+164:
		tmp = a
	elif t <= 9.5e+161:
		tmp = (z + a) - b
	else:
		tmp = a
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (t <= -3.5e+164)
		tmp = a;
	elseif (t <= 9.5e+161)
		tmp = Float64(Float64(z + a) - b);
	else
		tmp = a;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (t <= -3.5e+164)
		tmp = a;
	elseif (t <= 9.5e+161)
		tmp = (z + a) - b;
	else
		tmp = a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[t, -3.5e+164], a, If[LessEqual[t, 9.5e+161], N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision], a]]

Derivation

1. Split input into 2 regimes

2. if t < -3.4999999999999998e164 or 9.50000000000000061e161 < t

   1. Initial program 47.9%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in t around inf 55.4%

      \[\leadsto \color{blue}{a} \]

   if -3.4999999999999998e164 < t < 9.50000000000000061e161

   1. Initial program 60.8%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in y around inf 63.2%

      \[\leadsto \color{blue}{\left(a + z\right) - b} \]
   3. Step-by-step derivation

      1. +-commutative 63.2%

         \[\leadsto \color{blue}{\left(z + a\right)} - b \]

   4. Simplified 63.2%

      \[\leadsto \color{blue}{\left(z + a\right) - b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.5 \cdot 10^{+164}:\\ \;\;\;\;a\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+161}:\\ \;\;\;\;\left(z + a\right) - b\\ \mathbf{else}:\\ \;\;\;\;a\\ \end{array} \]

Alternative 13: 45.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-18}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3200:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (if (<= z -7.5e-18) z (if (<= z 3200.0) a z)))

C

double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7.5e-18) {
		tmp = z;
	} else if (z <= 3200.0) {
		tmp = a;
	} else {
		tmp = z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: tmp
    if (z <= (-7.5d-18)) then
        tmp = z
    else if (z <= 3200.0d0) then
        tmp = a
    else
        tmp = z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double tmp;
	if (z <= -7.5e-18) {
		tmp = z;
	} else if (z <= 3200.0) {
		tmp = a;
	} else {
		tmp = z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	tmp = 0
	if z <= -7.5e-18:
		tmp = z
	elif z <= 3200.0:
		tmp = a
	else:
		tmp = z
	return tmp

Julia

function code(x, y, z, t, a, b)
	tmp = 0.0
	if (z <= -7.5e-18)
		tmp = z;
	elseif (z <= 3200.0)
		tmp = a;
	else
		tmp = z;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	tmp = 0.0;
	if (z <= -7.5e-18)
		tmp = z;
	elseif (z <= 3200.0)
		tmp = a;
	else
		tmp = z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := If[LessEqual[z, -7.5e-18], z, If[LessEqual[z, 3200.0], a, z]]

Derivation

1. Split input into 2 regimes

2. if z < -7.50000000000000015e-18 or 3200 < z

   1. Initial program 51.0%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in x around inf 49.2%

      \[\leadsto \color{blue}{z} \]

   if -7.50000000000000015e-18 < z < 3200

   1. Initial program 67.8%

      \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

   2. Taylor expanded in t around inf 47.4%

      \[\leadsto \color{blue}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{-18}:\\ \;\;\;\;z\\ \mathbf{elif}\;z \leq 3200:\\ \;\;\;\;a\\ \mathbf{else}:\\ \;\;\;\;z\\ \end{array} \]

Alternative 14: 32.6% accurate, 21.0× speedup

Math

\[\begin{array}{l} \\ a \end{array} \]

FPCore

(FPCore (x y z t a b) :precision binary64 a)

C

double code(double x, double y, double z, double t, double a, double b) {
	return a;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    code = a
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return a;
}

Python

def code(x, y, z, t, a, b):
	return a

Julia

function code(x, y, z, t, a, b)
	return a
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = a;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := a

Derivation

1. Initial program 58.0%

   \[\frac{\left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b}{\left(x + t\right) + y} \]

2. Taylor expanded in t around inf 31.3%

   \[\leadsto \color{blue}{a} \]

3. Final simplification 31.3%

   \[\leadsto a \]

Developer target: 82.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + t\right) + y\\ t_2 := \left(\left(x + y\right) \cdot z + \left(t + y\right) \cdot a\right) - y \cdot b\\ t_3 := \frac{t_2}{t_1}\\ t_4 := \left(z + a\right) - b\\ \mathbf{if}\;t_3 < -3.5813117084150564 \cdot 10^{+153}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;t_3 < 1.2285964308315609 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{\frac{t_1}{t_2}}\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ (+ x t) y))
        (t_2 (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))
        (t_3 (/ t_2 t_1))
        (t_4 (- (+ z a) b)))
   (if (< t_3 -3.5813117084150564e+153)
     t_4
     (if (< t_3 1.2285964308315609e+82) (/ 1.0 (/ t_1 t_2)) t_4))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b);
	double t_3 = t_2 / t_1;
	double t_4 = (z + a) - b;
	double tmp;
	if (t_3 < -3.5813117084150564e+153) {
		tmp = t_4;
	} else if (t_3 < 1.2285964308315609e+82) {
		tmp = 1.0 / (t_1 / t_2);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_1 = (x + t) + y
    t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b)
    t_3 = t_2 / t_1
    t_4 = (z + a) - b
    if (t_3 < (-3.5813117084150564d+153)) then
        tmp = t_4
    else if (t_3 < 1.2285964308315609d+82) then
        tmp = 1.0d0 / (t_1 / t_2)
    else
        tmp = t_4
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (x + t) + y;
	double t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b);
	double t_3 = t_2 / t_1;
	double t_4 = (z + a) - b;
	double tmp;
	if (t_3 < -3.5813117084150564e+153) {
		tmp = t_4;
	} else if (t_3 < 1.2285964308315609e+82) {
		tmp = 1.0 / (t_1 / t_2);
	} else {
		tmp = t_4;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (x + t) + y
	t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b)
	t_3 = t_2 / t_1
	t_4 = (z + a) - b
	tmp = 0
	if t_3 < -3.5813117084150564e+153:
		tmp = t_4
	elif t_3 < 1.2285964308315609e+82:
		tmp = 1.0 / (t_1 / t_2)
	else:
		tmp = t_4
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(x + t) + y)
	t_2 = Float64(Float64(Float64(Float64(x + y) * z) + Float64(Float64(t + y) * a)) - Float64(y * b))
	t_3 = Float64(t_2 / t_1)
	t_4 = Float64(Float64(z + a) - b)
	tmp = 0.0
	if (t_3 < -3.5813117084150564e+153)
		tmp = t_4;
	elseif (t_3 < 1.2285964308315609e+82)
		tmp = Float64(1.0 / Float64(t_1 / t_2));
	else
		tmp = t_4;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (x + t) + y;
	t_2 = (((x + y) * z) + ((t + y) * a)) - (y * b);
	t_3 = t_2 / t_1;
	t_4 = (z + a) - b;
	tmp = 0.0;
	if (t_3 < -3.5813117084150564e+153)
		tmp = t_4;
	elseif (t_3 < 1.2285964308315609e+82)
		tmp = 1.0 / (t_1 / t_2);
	else
		tmp = t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(x + t), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x + y), $MachinePrecision] * z), $MachinePrecision] + N[(N[(t + y), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision] - N[(y * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(N[(z + a), $MachinePrecision] - b), $MachinePrecision]}, If[Less[t$95$3, -3.5813117084150564e+153], t$95$4, If[Less[t$95$3, 1.2285964308315609e+82], N[(1.0 / N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]

Reproduce

herbie shell --seed 2023173 
(FPCore (x y z t a b)
  :name "AI.Clustering.Hierarchical.Internal:ward from clustering-0.2.1"
  :precision binary64

  :herbie-target
  (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) -3.5813117084150564e+153) (- (+ z a) b) (if (< (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)) 1.2285964308315609e+82) (/ 1.0 (/ (+ (+ x t) y) (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)))) (- (+ z a) b)))

  (/ (- (+ (* (+ x y) z) (* (+ t y) a)) (* y b)) (+ (+ x t) y)))